6 Multiscale entropy applied to background fluctuation analysis

The mean entropy vector

The multiscale entropy has been defined by:

$$H(X) = \sum_{j=1}^{l} \sum_{k=1}^{N} h(w_j)$$ (55)

with $h(w_j = \ln ( p(w_j(k))))$. In order to study the behavior of the information at a given scale, we prefer to calculate the mean entropy vector E defined by:

$$E(j) = \frac{1}{N}\sum_{k=1}^{N} h(w_j)$$ (56)

E(j) gives the mean entropy at scale j. From the mean entropy vector, we have statistical information on each scale separately. Having a noise model, we are able to calculate (generally from simulations) the mean entropy vector $E^{(\rm noise)}(j)$ resulting from pure noise. Then we define the normalized mean entropy vector by

$$E_n(j) = \frac{ E(j) }{ E^{(\rm noise)}(j) }\cdot$$ (57)

Figure 9 shows the result of a simulation.

Figure 9: Mean entropy versus the scale of 5 simulated images containing undetectable sources and noise. Each curve corresponds to the multiscale transform of one image. From top to bottom, the image contains respectively 400, 200, 100, 50 and 0 sources

Five simulated images were created by adding n sources to a $1024\times 1024$ image containing Gaussian noise of standard deviation equal to 1 (and arbitrary mean). The n sources are identical, with a maximum equal to 1, standard deviation equal to 2, and zero covariance terms. Defining the signal-to-noise ratio (SNR) as the ratio between the standard deviation in the smallest box which contains at least 90% of the flux of the source, and the noise standard deviation, we have a SNR equal to 0.25. The sources are not detectable in the simulated image, nor in its wavelet transform. Figure 10 shows a region which contains a source at the center. It is clear there is no way to find this kind of noisy signal. The five images were created using a number of sources respectively equal to 0, 50, 100, 200 and 400, and the simulation was repeated ten times with different noise maps in order to have an error bar on each entropy measurement. For the image which contains 400 sources, the number of pixels affected by a source is less than 2.5%.

When the number of sources increases, the difference between the multiscale entropy curves increases. Even if the sources are very faint, the presence of signal can be clearly detected using the mean entropy vector. But it is obvious that the positions of these sources remain unknown.